Generalized symmetric group

In mathematics, the generalized symmetric group is the wreath product ${\displaystyle S(m,n):=Z_{m}\wr S_{n}}$ of the cyclic group of order m and the symmetric group of order n.

Examples

• For ${\displaystyle m=1,}$ the generalized symmetric group is exactly the ordinary symmetric group: ${\displaystyle S(1,n)=S_{n}.}$
• For ${\displaystyle m=2,}$ one can consider the cyclic group of order 2 as positives and negatives (${\displaystyle Z_{2}\cong \{\pm 1\}}$) and identify the generalized symmetric group ${\displaystyle S(2,n)}$ with the signed symmetric group.

Representation theory

There is a natural representation of elements of ${\displaystyle S(m,n)}$ as generalized permutation matrices, where the nonzero entries are m-th roots of unity: ${\displaystyle Z_{m}\cong \mu _{m}.}$

The representation theory has been studied since (Osima 1954); see references in (Can 1996). As with the symmetric group, the representations can be constructed in terms of Specht modules; see (Can 1996).

Homology

The first group homology group (concretely, the abelianization) is ${\displaystyle Z_{m}\times Z_{2}}$ (for m odd this is isomorphic to ${\displaystyle Z_{2m}}$): the ${\displaystyle Z_{m}}$ factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to ${\displaystyle Z_{m}}$ (concretely, by taking the product of all the ${\displaystyle Z_{m}}$ values), while the sign map on the symmetric group yields the ${\displaystyle Z_{2}.}$ These are independent, and generate the group, hence are the abelianization.

The second homology group (in classical terms, the Schur multiplier) is given by (Davies & Morris 1974):

${\displaystyle H_{2}(S(2k+1,n))={\begin{cases}1&n<4\\\mathbf {Z} /2&n\geq 4.\end{cases}}}$

${\displaystyle H_{2}(S(2k+2,n))={\begin{cases}1&n=0,1\\\mathbf {Z} /2&n=2\\(\mathbf {Z} /2)^{2}&n=3\\(\mathbf {Z} /2)^{3}&n\geq 4.\end{cases}}}$

Note that it depends on n and the parity of m: ${\displaystyle H_{2}(S(2k+1,n))\approx H_{2}(S(1,n))}$ and ${\displaystyle H_{2}(S(2k+2,n))\approx H_{2}(S(2,n)),}$ which are the Schur multipliers of the symmetric group and signed symmetric group.

References